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3 years ago

How do you find the value of a number

1 answer:

7 0

We calculate it by multiplying the place value and face value of the digit. For instance: If we consider a number 45. Here the digit 4 is in the tens column.

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Let f(x,y,z) = ztan-1(y2) i + z3ln(x2 + 1) j + z k. find the flux of f across the part of the paraboloid x2 + y2 + z = 3 that li

Sophie [7]

Consider the closed region $V$ bounded simultaneously by the paraboloid and plane, jointly denoted $S$ . By the divergence theorem,

$\displaystyle\iint_S\mathbf f(x,y,z)\cdot\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

And since we have

$\nabla\cdot\mathbf f(x,y,z)=1$

the volume integral will be much easier to compute. Converting to cylindrical coordinates, we have

$\displaystyle\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\iiint_V\mathrm dV$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}\int_{z=2}^{z=3-r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=1}r(3-r^2-2)\,\mathrm dr$
$=\dfrac\pi2$

Then the integral over the paraboloid would be the difference of the integral over the total surface and the integral over the disk. Denoting the disk by $D$ , we have

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-\iint_D\mathbf f\cdot\mathrm dS$

Parameterize $D$ by

$\mathbf s(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+2\,\mathbf k$
$\implies\mathbf s_u\times\mathbf s_v=u\,\mathbf k$

which would give a unit normal vector of $\mathbf k$ . However, the divergence theorem requires that the closed surface $S$ be oriented with outward-pointing normal vectors, which means we should instead use $\mathbf s_v\times\mathbf s_u=-u\,\mathbf k$ .

Now,

$\displaystyle\iint_D\mathbf f\cdot\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(-u\,\mathbf k)\,\mathrm dv\,\mathrm du$
$=\displaystyle-4\pi\int_{u=0}^{u=1}u\,\mathrm du$
$=-2\pi$

So, the flux over the paraboloid alone is

$\displaystyle\iint_{S-D}\mathbf f\cdot\mathrm dS=\frac\pi2-(-2\pi)=\dfrac{5\pi}2$

6 0

4 years ago

Every minute, 2.2gallons of water flows from a shower. A family of 5 people showers for an average of 9 minutes per person every

satela [25.4K]

Answer:

the whole family uses a total of 99 gallons in 1 morning

Step-by-step explanation:

7 0

3 years ago

A boat captain can see a lighthouse on the shore which is 103ft in front of him. If the lighthouse is 22ft tall, at what angle o

elena-14-01-66 [18.8K]

Answer:

21°

Step-by-step explanation:

A boat captain, lighthouse and its base form a right triangle.

<u>We need to find the angle opposite to 22 ft side:</u>

tan x = 22/103
x = arctan (22/103)
x = 21°

<em>Note, option D should read 21° not 11°</em>

5 0

2 years ago

The systolic blood pressures of women aged 18 to 24 are normally distributed with a mean of 115.3 and a standard deviation of 13

Ghella [55]

Answer:

fac.k nou ok your mather xd

Step-by-step explanation:

6 0

3 years ago

I need help please right answers only

yarga [219]

Answer:

-3/2

Step-by-step explanation:

Count up three times and to the left twice. Let me know if you need help with other questions!

7 0

3 years ago

How do you find the value of a number ​

How do you find the value of a number